Lyapunov Inequalities Of Dynamic Equations On Time Scales

The theory of dynamic equations on time scale was initiated by Hilger in 1990 in order to create a theory that can unify discrete and continuous analysis. During the last few years, some Lyapunov inequalities for dynamic equations on time scales have been obtained by many authors. In this thesis we study some Lyapunov inequalities of dynamic equations (or systems) on time scales.In Chapter one, we introduce the basc theory of dynamic equations and the current situation for the Lyapunov inequalities of dynamic equations.In Chapter two and three, we investigate Hamiltonian systems xΔ(t)=A(t)x(a(t))-B(t)y(t), yΔ(t)＝C(t)x(σ(t))＋AT(t)Y(t), and quasi-Hamiltonian systems =-A(t)x(σ(t))-B(t)|y(t)|p-2y(t), = C(t)|x(σ)|q-2x(σ(t))+AT(t)y(t), on time scale T respectively, and obtain several Lyapunov inequalities of those systems under some conditions, where p, q ∈ (1,+∞) satisfying 1/p+1/q= 1, A(t) is a real n×n matrix-valued function on T and I+μ(t)A(t) is invertible, B(t) and C(t) are two real n×n symmetric matrix-valued functions on T and B(t) is positive definite, and x(t),y(t) are two real n-dimensional vector-valued functions on T.In Chapter four, we investigate the following higher-order dynamic equation SnΔ(t,x(t)+φ(t)(t)=0 on time scale T, and obtain some Lyapunov inequalities of that equation under some con-ditions, where n is a positive integer, β(≥1) is a quotient of two odd positive integers, S0(t,x(t))= x(t) (k= 0), Sk(t,x(t))=ak(t)Sk-1Δ(t,x(t) (1≤k≤n-1), Sn(t,x(t))= an(t)[Sn-1Δ(t,x(t_)]β, and ak ∈ Crd(T, (0,∞>)) 1≤k≤n),φ(t)∈Crd(T,R).In Chapter five, we investigate the following higher-order dynamic equation |SnΔ(t,X(t))|p-2SnΔ(t,X(t))+B(t)|X(t)p-2X(t)=0 on time scale T, and obtain a Lyapunov inequality of that equation under anti-periodic bound-ary conditions, where n is a positive integer, p ∈ (1,+∞), X(t) is a rea n-dimensional vector-valued functions on T, S0(t,X(t))= X(t) and Sk(t,X(t))= Ak(t)Sk-1Δ(t,X(t))(1≤k≤n), Ak(t) (1≤k≤n) are real n×n positive definite matrix-valued functions on T and B(t) is a real n×n matrix-valued function on T with I+μ(t)B(t) being invertible.